Eshelby’s circular cylindrical inclusion with polynomial eigenstrains in transverse direction by residue theorem

Cylindrical inclusions with constant cross section in an infinite isotropic matrix are usually treated as plane elasticity problems and solved by complex potential method without considering the longitudinal eigenstrains. This paper provides a closed-form solution for the Eshelby’s circular cylindrical inclusion with eigenstrains which are polynomial in transverse direction and uniform in longitudinal direction. The integrals of Green’s function are decomposed into the sum of customized L-integrals. Two sets of L-integrals for the regions inside and outside the circular cylindrical inclusion are evaluated by using the residue theorem. Further, the stress and strain fields inside and outside the inclusion resulted from the polynomial eigenstrains are obtained. Circular cylinder inclusions with uniform, linear, and quadric eigenstrains are, respectively, used as examples to illustrate the proposed solution. When the cylindrical inclusion only suffers transverse eigenstrains, the solution is appropriate for the circular inclusion with polynomial eigenstrains in plane elasticity. The proposed method has convenient formulae and simplifies the integrals of Green’s function with polynomial eigenstrains.